GroupJive Groups have the following types in addition to access (usergroup based) permission.
Public - Can be joined by anyone, data visibility is based off "Public" parameters of integration tabs.. so can be Publicly joined, but content is private to the group, etc..
Approval - Can be viewed and joined, but join needs to be approved by group owner, group admin, or group moderator... see Public description for data public display.
Invite Only - Can not be viewed or joined unless the user is invited by someone in the group.
For users the only solution we've available for privacy control is CB Privacy, which lets the user set who can see the tabs/fields on their profile. This is not usergroup based.
Media, I've no idea your exact usage but GroupJive comes with a Video, Event, Photo, Wall, File, and more integrations that are Group by Group based. We've no dedicated solution for media except CB ProfileGallery, which only does files and videos and has no relation to GroupJive.